AREA OF EQUILATERAL TRIANGLE

What is equilateral triangle ?

In geometry, an equilateral triangle is a triangle in which all three sides are equal. The area enclosed by this shape is known as area of equilateral triangle.

Area of equilateral triangle can be found using the formula given below.

Area of Equilateral Triangle  =  (√3/4)a² sq. units

where a is the length of each side of the triangle. 

Deriving the Formula to Find the Area of Equilateral Triangle

Take an equilateral triangle of the side “a” units. Then draw a perpendicular bisector to the base of height “h”.

By drawing perpendicular from A, we get two congurent right triangle ABD and ADC.

Area of triangle ABC  =  Area of triangle ABD + Area of triangle ADC

Since triangles ABD and ADC are congurent, areas will be equal.

Area of triangle ABC  =  Area of ABD + Area of ADC

Area of triangle ABC  =  2  (Area of ABD)

=  2 ⋅ (1/2) ⋅ Base ⋅ Height

Area of triangle ABC  =  Base ⋅ Height  ---(1)

In triangle ABD,

Base (BD)  =  a/2 and height (AD)  =  h

Using Pythagorean theorem,

a²  =  h² + (a/2)²

h²  =  a² - (a²/4)

h²  =  (3a²/4)

h  =  √(3a²/4)

h  =  (a√3/2)

By applying the values of base and height in (1), we get

Area of triangle ABC  =  (a/2) ⋅ (a√3/2)

  =  (1/4)a²√3

Area of triangle ABC  =  √3a²/4 square units

Examples

Example 1 :

Find the area of the equilateral triangle having the length of the side equals 10 cm.

Solution :

Area of equilateral triangle  =  (√3/4) a²

Here a = 10 cm

  =  (√3/4) (10)²

  =  (√3/4) x (10) x (10)

  =  (√3) x (5) x (5)

  =  25 √3 cm²

Example 2 :

Find the length of the altitude of an equilateral triangle of side 3√3 cm.

Solution :

Side length of equilateral triangle (a)  =  3√3

Area of equilateral triangle  =  (√3/4) a²

  =  (√3/4) (3√3)²

=  (√3/4) (27)

Area of equilateral triangle  =  27√3 / 4   ---(1)

Here we should find the length of altitude,  so we use the formula base ⋅ height to find the area of equilateral triangle.

=   base ⋅ height

  =  (3√3/2) ⋅ h    ------(2)

(1)  =  (2)

(3√3/2) ⋅ h  =  27√3/4

h  =  (27√3/4) ⋅ (2/3√3)

h  =  9/2 

h  =  4.5

So, the required height is 4.5 cm.

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